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Numerous theoretical studies using various equation of state models haveshown that quark matter may exist at the extreme densities in the cores ofhigh-mass neutron stars. It has also been shown that a phase transition fromhadronic matter to quark matter would result in an extended mixed phase regionthat would segregate phases by net charge to minimize the total energy of thephase, leading to the formation of a crystalline lattice. The existence ofquark matter in the core of a neutron star may have significant consequencesfor its thermal evolution, which for thousands of years is facilitatedprimarily by neutrino emission. In this work we investigate the effect acrystalline quark-hadron mixed phase can have on the neutrino emissivity fromthe core. To this end we calculate the equation of state using the relativisticmean-field approximation to model hadronic matter and a nonlocal extension ofthe three-flavor Nambu-Jona-Lasinio model for quark matter. Next we determinethe extent of the quark-hadron mixed phase and its crystalline structure usingthe Glendenning construction, allowing for the formation of spherical blob,rod, and slab rare phase geometries. Finally we calculate the neutrinoemissivity due to electron-lattice interactions utilizing the formalismdeveloped for the analogous process in neutron star crusts. We find that thecontribution to the neutrino emissivity due to the presence of a crystallinequark-hadron mixed phase is substantial compared to other mechanisms at fairlylow temperatures ($\lesssim 10^9$ K) and quark fractions ($\lesssim 30\%$), andthat contributions due to lattice vibrations are insignificant compared tostatic-lattice contributions.

Neutrino emissivity in the quark-hadron mixed phase of neutron stars